Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 35, 2010, 473-492

BOUNDED DISTORTION HOMEOMORPHISMS ON ULTRAMETRIC SPACES

Bruce Hughes, Álvaro Martínez-Pérez and Manuel A. Morón

Vanderbilt University, Department of Mathematics
1326 Stevenson Center, Nashville, TN 37240, U.S.A.; bruce.hughes 'at' vanderbilt.edu

Universidad Complutense de Madrid, Departamento de Geometría y Topología
Plaza de Ciencias 3, Madrid 28040, Spain; alvaro_martinez 'at' mat.ucm.es

Universidad Complutense de Madrid, Departamento de Geometría y Topología
Plaza de Ciencias 3, Madrid 28040, Spain; mamoron 'at' mat.ucm.es

Abstract. It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity as the end spaces of bushy trees). A bounded distortion property is found that characterizes power quasi-symmetric homeomorphisms between such ultrametric spaces that are also pseudo-doubling. Moreover, examples are given showing the extent to which the power quasi-symmetry of homeomorphisms is not captured by the quasiconformal and bi-Hölder conditions for this class of ultrametric spaces.

2000 Mathematics Subject Classification: Primary 54E40, 30C65, 53C23.

Key words: Tree, real tree, bushy tree, ultrametric, end space, quasi-isometry, quasiconformal, quasi-symmetric, PQ-symmetric, doubling metric space.

Reference to this article: B. Hughes, Á. Martínez-Pérez and M.A. Morón: Bounded distortion homeomorphisms on ultrametric spaces. Ann. Acad. Sci. Fenn. Math. 35 (2010), 473-492.

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doi:10.5186/aasfm.2010.3529